Ivory's Theorem revisited

TL;DR

This paper revisits Ivory's Lemma, deriving it from integrability of billiards, explores its analogs in various metrics, and extends classical potential theorems to curved spaces, revealing deep geometric and physical insights.

Contribution

It provides a new derivation of Ivory's Lemma from billiard integrability, explores its generalizations in different geometries, and extends classical potential theorems to curved spaces.

Findings

Ivory's Lemma derived from billiard integrability

Generalizations of Ivory's Lemma in Liouville and Stäckel metrics

Extension of potential theorems to spherical and hyperbolic spaces

Abstract

Ivory's Lemma is a geometrical statement in the heart of J. Ivory's calculation of the gravitational potential of a homeoidal shell. In the simplest planar case, it claims that the diagonals of a curvilinear quadrilateral made by arcs of confocal ellipses and hyperbolas are equal. In the first part of this paper, we deduce Ivory's Lemma and its numerous generalizations from complete integrability of billiards on conics and quadrics. In the second part, we study analogs of Ivory's Lemma in Liouville and St\"ackel metrics. Our main focus is on the results of the German school of differential geometry obtained in the late 19 -- early 20th centuries that might be lesser know today. In the third part, we generalize Newton's, Laplace's, and Ivory's theorems on gravitational and Coulomb potential of spheres and ellipsoids to the spherical and hyperbolic spaces. V. Arnold extended the results of Newton, Laplace, and Ivory to algebraic hypersurfaces in Euclidean space; we generalize Arnold's theorem to the spaces of constant curvature.